Question

If $AM:GM=5:4$ for two positive numbers then the ratio of the numbers is

• A. $4:1$
• B. $4:3$
• C. $2:3$
• D. $Noneofthese$

Answer 1

The correct option is A $4:1$

$\frac{AM}{GM}=\frac{5}{4}$

$\Rightarrow \frac{a+b/2}{\sqrt{ab}}=\frac{5}{4}$

$AM=\frac{a+b}{2}$ & $GM=\sqrt{ab}$

$\Rightarrow \frac{a+b}{2\left(\sqrt{ab}\right)}=\frac{5}{4}$

$\Rightarrow \frac{a+b}{\sqrt{ab}}=\frac{5}{2}$ squaring both sides

$\Rightarrow \frac{(a+b{)}^2}{(\sqrt{ab}{)}^2}={\left(\frac{5}{2}\right)}^2$

$\Rightarrow \frac{a^2+b^2+2ab}{ab}=\frac{25}{4}$

$\frac{a}{b}+\frac{b}{a}+2=\frac{25}{4}$

$m+\frac{1}{m}=\frac{25}{4}=2$

$\left\{let\begin{array}{cc}\frac{a}{b}=m,& so\\ \frac{b}{a}& \frac{1}{m}\end{array}\right\}$.